HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslj1i Unicode version

Theorem mdslj1i 23013
Description: Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslle1.1  |-  A  e. 
CH
mdslle1.2  |-  B  e. 
CH
mdslle1.3  |-  C  e. 
CH
mdslle1.4  |-  D  e. 
CH
Assertion
Ref Expression
mdslj1i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )

Proof of Theorem mdslj1i
StepHypRef Expression
1 ssin 3467 . . . . 5  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
21bicomi 193 . . . 4  |-  ( A 
C_  ( C  i^i  D )  <->  ( A  C_  C  /\  A  C_  D
) )
3 mdslle1.3 . . . . . 6  |-  C  e. 
CH
4 mdslle1.4 . . . . . 6  |-  D  e. 
CH
5 mdslle1.1 . . . . . . 7  |-  A  e. 
CH
6 mdslle1.2 . . . . . . 7  |-  B  e. 
CH
75, 6chjcli 22150 . . . . . 6  |-  ( A  vH  B )  e. 
CH
83, 4, 7chlubi 22164 . . . . 5  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
98bicomi 193 . . . 4  |-  ( ( C  vH  D ) 
C_  ( A  vH  B )  <->  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )
102, 9anbi12i 678 . . 3  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  <->  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )
11 simpr 447 . . . . . . . . . 10  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
12 simpl 443 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  C )
13 simpl 443 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  C  C_  ( A  vH  B
) )
145, 6, 33pm3.2i 1130 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
15 dmdsl3 23009 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
1614, 15mpan 651 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  A )  =  C )
1711, 12, 13, 16syl3an 1224 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
183, 6chincli 22153 . . . . . . . . . . 11  |-  ( C  i^i  B )  e. 
CH
194, 6chincli 22153 . . . . . . . . . . 11  |-  ( D  i^i  B )  e. 
CH
2018, 19chub1i 22162 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
2118, 19chjcli 22150 . . . . . . . . . . 11  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH
2218, 21, 5chlej1i 22166 . . . . . . . . . 10  |-  ( ( C  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2320, 22mp1i 11 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2417, 23eqsstr3d 3289 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  C  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
25 simpr 447 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
26 simpr 447 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
275, 6, 43pm3.2i 1130 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
28 dmdsl3 23009 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
2927, 28mpan 651 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
3011, 25, 26, 29syl3an 1224 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
3119, 18chub2i 22163 . . . . . . . . . 10  |-  ( D  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
3219, 21, 5chlej1i 22166 . . . . . . . . . 10  |-  ( ( D  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3331, 32mp1i 11 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3430, 33eqsstr3d 3289 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3524, 34jca 518 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) ) )
3621, 5chjcli 22150 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  e.  CH
373, 4, 36chlubi 22164 . . . . . . 7  |-  ( ( C  C_  ( (
( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )  <->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3835, 37sylib 188 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
39 ssrin 3470 . . . . . 6  |-  ( ( C  vH  D ) 
C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A
)  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
4038, 39syl 15 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
41 simpl 443 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
42 ssrin 3470 . . . . . . . 8  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( C  i^i  B ) )
4342, 20syl6ss 3267 . . . . . . 7  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
4443adantr 451 . . . . . 6  |-  ( ( A  C_  C  /\  A  C_  D )  -> 
( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
45 inss2 3466 . . . . . . . 8  |-  ( C  i^i  B )  C_  B
46 inss2 3466 . . . . . . . 8  |-  ( D  i^i  B )  C_  B
4718, 19, 6chlubi 22164 . . . . . . . . 9  |-  ( ( ( C  i^i  B
)  C_  B  /\  ( D  i^i  B ) 
C_  B )  <->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B )
4847bicomi 193 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B 
<->  ( ( C  i^i  B )  C_  B  /\  ( D  i^i  B ) 
C_  B ) )
4945, 46, 48mpbir2an 886 . . . . . . 7  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
5049a1i 10 . . . . . 6  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B )
515, 6, 213pm3.2i 1130 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )
52 mdsl3 23010 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
) )  ->  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
5351, 52mpan 651 . . . . . 6  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
)  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5441, 44, 50, 53syl3an 1224 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5540, 54sseqtrd 3290 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
56553expb 1152 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5710, 56sylan2b 461 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
583, 4, 6lediri 22230 . . 3  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  (
( C  vH  D
)  i^i  B )
5958a1i 10 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  ( ( C  vH  D )  i^i 
B ) )
6057, 59eqssd 3272 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   class class class wbr 4104  (class class class)co 5945   CHcch 21623    vH chj 21627    MH cmd 21660    MH* cdmd 21661
This theorem is referenced by:  mdslmd1lem1  23019  mdslmd1lem2  23020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cc 8151  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907  ax-hilex 21693  ax-hfvadd 21694  ax-hvcom 21695  ax-hvass 21696  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvmulass 21701  ax-hvdistr1 21702  ax-hvdistr2 21703  ax-hvmul0 21704  ax-hfi 21772  ax-his1 21775  ax-his2 21776  ax-his3 21777  ax-his4 21778  ax-hcompl 21895
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-omul 6571  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-acn 7665  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-rlim 12059  df-sum 12256  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-cn 17063  df-cnp 17064  df-lm 17065  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cfil 18785  df-cau 18786  df-cmet 18787  df-grpo 20970  df-gid 20971  df-ginv 20972  df-gdiv 20973  df-ablo 21061  df-subgo 21081  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-vs 21269  df-nmcv 21270  df-ims 21271  df-dip 21388  df-ssp 21412  df-ph 21505  df-cbn 21556  df-hnorm 21662  df-hba 21663  df-hvsub 21665  df-hlim 21666  df-hcau 21667  df-sh 21900  df-ch 21915  df-oc 21945  df-ch0 21946  df-shs 22001  df-chj 22003  df-md 22974  df-dmd 22975
  Copyright terms: Public domain W3C validator